# Is a negative number rational

### Natural numbers \$\$ NN \$\$

The number range of the natural numbers \$\$ NN \$\$ forms that counting as a natural process.

• The smallest natural number is the \$\$ 0 \$\$.

• The set of natural numbers contains all successors of the \$\$ 0 \$\$ up to infinity:
\$\$ NN = {0,1,2,3,4, ..., n, n + 1, ...} \$\$ .

### How can you calculate with natural numbers?

You are allowed without restriction add and multiply.

• It is said that \$\$ NN \$\$ is related to addition and multiplication completed.
• All other arithmetic operations cannot be carried out without restrictions.

### Whole numbers \$\$ ZZ \$\$

If you expand the number range of the natural numbers with the negative numbers, do you have the whole numbers:

• In the set of negative numbers are all positive and negative numbers without comma: \$\$ ZZ = {…, -3, -2, -1,0,1,2,3,…} \$\$
• Now you can also without restrictions subtract.

Successor principle: Is \$\$ n \$\$ is any natural number, then \$\$ n + 1 \$\$ her successor.

Example: The number \$\$ n = 73 \$\$ has the successor \$\$ n + 1 = 74 \$\$

Seclusion: The result of the calculation is the same amount, here \$\$ NN \$\$.

Example:

• If you add two natural numbers, the sum is also a natural number. \$\$ 4 + 3 = 7 \$\$
• If you calculate \$\$ 4: 3 \$\$, the result is not a natural number, but a fraction \$\$4/3\$\$.

### Broken Numbers \$\$ QQ \$\$\$\$+\$\$

Do you want unlimited to divide, you need the fractions.

• \$\$ QQ \$\$\$\$+\$\$ contains all positive fractions
• \$\$ QQ \$\$\$\$+\$\$\$\$ = \$\$\$\$ {\$\$\$\$ a / b | \$\$ \$\$ a, b \$\$ is a natural number and \$\$ b! = 0} \$\$

### Rational Numbers \$\$ QQ \$\$

Do you take the negative fractions in addition, you have the rational numbers.

• \$\$ QQ = {\$\$\$\$ a / b | \$\$ \$\$ a \$\$ is an integer \$\$, b \$\$ is a natural number and \$\$ b! = 0} \$\$
• In \$\$ QQ \$\$ you can all basic arithmetic run without restriction.
• \$\$ QQ \$\$ contains all positive and negative fractions, as well as all terminating Decimal fractions (e.g. \$\$ - 3.75 \$\$) and periodic Decimal fractions (e.g. \$\$ 0.66666 ... \$\$).
##### \$\$ a \$\$ can be negative, so the quotient can also be negative.

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### Number ranges in one another

The number ranges lie one inside the other.

For example, the \$\$ 2 \$\$ is a natural number, an integer, and a rational number. 